5.5. FREQUENCY DISPERSION OF MULTILAYER MICROSTRIP LINES 187
e last layer q
N
is again given by Eq. (5.11d).
Based on the transformed domain, (Figure 5.3b), Svacina extracted a general expression
for the effective dielectric constant as:
"
reff
D
P
M
iD1
q
i
2
P
M
iD1
q
i
"
ri
2
C
P
N
iDMC1
q
j
2
P
N
j DM C1
q
j
"
rj
2
: (5.14)
Numerous authors verified the above expression with accuracy better than 2%. In contrary
Wan and Hoorfar [13] made a comparison against a moment method solution for a microstrip
located on the first layer and for numbers of layers more than three. ey found a large inaccuracy
the order of 14–35% and suggested an improvement. However, a careful examination of [13]
reveals that the authors misinterpreted the indeed difficult-to-read format of Svacina [12] ex-
pressions. Actually, the improvement suggested in [13] was already there, hidden in the [12]
expressions.
Characteristic Impedance
A relatively simple formula represents Z
0
in terms of the effective dielectric constant as [3]:
Z
0
.f ! 0/ D
Z
0;air
p
"
reff
.f D 0/
; (5.15a)
where Z
0;air
is the characteristic impedance of the air-filled ."
r
D
r
D 1/ microstrip:
Z
0;air
D
(
127:415 57:839 ln
.
Nw
/
for 0:05 Nw 2
136:6131 30:88 Nw C 33:43 Nw
2
0:13384 Nw
3
for 2 Nw 10
)
: (5.15b)
An alternative expression for Z
0;air
D Z
LO
is given in [3] which is originally proposed by
Hammerstad and Jensen [14].
5.5 FREQUENCY DISPERSION OF MULTILAYER
MICROSTRIP LINES
e effective dielectric constant given in the previous sections is based on a conformal transfor-
mation which is exactly valid in the static case (f D 0 Hz), to be symbolized as "
reff
.f D 0/.
rough a quasi-static approach, the above has an acceptable accuracy for low microwave fre-
quencies .f ! 0/. However, microstrip lines are frequency dispersive and their effective dielec-
tric constant depends on frequency, "
reff
.f /. Verma and Sadr [15] proposed a dispersion model
for multilayer structures by adapting the single-layer model of Kirschning and Jansen [16], which
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